dept. of computer science & it, fuuast automata theory 2 automata theory iii languages and...

AUTOMATATHEORY

III

Dept. of Computer Science & IT, FUUAST Automata Theory 2

Automata Theory III

Languages And Regular Expressions

Construction of FA’s for given languages

S = { 0,1 }L0,L1, L2, L3, …………Ln L* = L0 L1 L2 L3 ……… Ln

3Dept. of Computer Science & IT, FUUAST Automata Theory

Automata Theory III

L = {w | bm abn : m, n0}

= {a, b }


Automata Theory III

L = {w | am bn : m, n0}

= {a, b }


Automata Theory III

= {a, b }

L={ w| all strings with the prefix ‘ab’ on


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= {0, 1}

L={0,1}* , having even number of 0’s


Automata Theory III

= {0, 1}

L={w {0,1}* | having odd number of 0’s}


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= {a, b }L={ (ab)*| n0 }, not accepted.


Automata Theory III

= {a, b }L={ (ab)*| n1 }, not accepted.


Automata Theory III

= {0,1,… 9}

L= set of all integers


Automata Theory III

= {+,-,0,1,… 9}

L= set of all signed integers


Automata Theory III

= {a, b }L={ an| n0 } { bna | n 1}


Automata Theory III

Regular Expressions ( Regex )

import java.util.regex.*; ‘Pattern’ and ‘Matcher’ classesusing System.Text.RegularExpressions;

Java:

C#:

‘Regex’ class

1) Wild Cards2) Computer Languages

C++, VB, Perl, Html, Oracle etc………….3) Operating Systems

Unix, Linux, Windows, etc. (grep, ed, awk, vi, expr)

Searching and Validation

A regular expression, often called a pattern, is an expression that describes a set of strings. They are usually used to give a concise description of a set, without having to list all elements.


Automata Theory III

Examples of Regular Expressions:1) Email Address:

\b[A-Z0-9._%+-]+@[A-Z0-9.-]+\.[A-Z]{2,4}\b2) IP Address:

\b(25[0-5]|2[0-4][0-9]|[01]?[0-9][0-9]?)\.(25[0-5]|2[0-4][0-9]|[01]?[0-9][0-9]?)\.(25[0-5]|2[0-4][0-9]|[01]?[0-9][0-9]?)\.(25[0-5]|2[0-4][0-9]|[01]?[0-9][0-9]?)\b

3) Valid Date:(19|20)\d\d[- /.](0[1-9]|1[012])[- /.](0[1-9]|[12][0-9]|3[01])

4) Visa Card: ^4[0-9]{12}(?:[0-9]{3})?$

5) Master Card: ^5[1-5][0-9]{14}$

6) Strings:"[^"\r\n]*"

7) Numbers:\b\d+\b, \b0[xX][0-9a-fA-F]+\b, (\b[0-9]+\.([0-9]+\b)?|\.[0-

9]+\b)


Automata Theory III

Representation of Regular ExpressionsIt is obtained from a set of symbols , empty string , null set , performing operations, union(+), concatenation(.), and Kleene star(*). Regular Expression defines Regular Languages as automata do. L(E) is a language defined by regular expression ‘E’

BASIS: 1) and are regular expressions, L()={} and L() ={} are corresponding regular Languages. 2) ‘a’ is a symbol, then a is a regular expression and L(a) ={a} is corresponding regular language.


Automata Theory III

INDUCTION:

Precedence of Regular Expression Operators:1) *2) .3) +


Automata Theory III

If P, Q and R are regular expressions; then

1) R + R = R2) + R = R3) R = R = 4) R = R = R5) R* = + R + R2 + R3 + R4 + R5 + . . . . . .6) (PQ)R = P(QR)7) PQ QP8) P (Q + R) = PQ + PR9) (P + R)Q = PQ + RQ10) R*R* = R*11) RR* = R*R12) (R*)* = R*13) (P + Q)* = (P*Q*)* = (P* + Q*)*


Automata Theory III

Examples:1) 0 + 12) 13) 04) (0 + 1)15) (a + b).(b + c)6) (0+1)*7) (0+1)+

8) ( + 1)(01)*( + 0)


Automata Theory III

SET Regular Expression1) { ^, ab }2) {1,11,111, ….. }3) Set of all strings over {a,b} beginning and ending with ‘a’4) {b2, b5, b8, ……. }5) {a2n+1| n0}6) {a2nb2m+1| n0}7) Strings over {a,b} beginning with 0 and ending with 1

^ + ab 1(1)*a (a + b)*a

bb(bbb)*a (aa)*(aa)*(bb)*b0 (0 + 1)*1


Automata Theory III

Def.:

If is set of alphabet:1) is a regular language2) For each a , {a} is a regular language3) If L0, L1, L2, L3, …….. Ln are regular languages then

L0 L1 L2 L3 ……. Ln is also regular language.4) If L0, L1, L2, L3, …….. Ln are regular languages then

L0 . L1 . L2 . L3 ……. . Ln is also regular language.5) If L is a regular language, then so is L*.


Automata Theory III

Finite Automata and Regular Expressions


Automata Theory III

From DFA To Regular Expressions


Automata Theory III


Automata Theory III

Converting DFA to Regular Expressions by Eliminating States

A generic two-state Automaton

A generic one-state Automaton


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EliminatingB and C


Automata Theory III


Automata Theory III

L(ε) = {ε}

L(Ф) = Ф

L(a) = {a}

ε- NFA


Automata Theory III

L = L(R) L(S)

L = L(R)L(S)

L = L(R*)


Automata Theory III

Conversion of Regular Expression (0 + 1)*1(0 + 1) to ε- NFA


Automata Theory III

THE ENDRegular Expressions (Chapter 3 )

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